Spiral galaxies are typically known to have three main components: a flattened disk, a bright central bulge with a dense concentration of stars, and an extended spherical halo of sparsely distributed stars. The galactic halos also host a large number of globular clusters.

Around 2004-2007, astronomers have found that Andromeda (M31), the nearest large spiral galaxy to the Milky Way, has an enormous stellar halo, much bigger than previously thought. It extends far beyond the swirling disk seen in photographic images. After this surprising discovery, kind of an explosion of scientific papers and press releases took place. It is now usually assumed that such big halos with radii of 150-200 kpc are a generic feature of (spiral?) galaxies!

Here is a nice illustration from Space.com of these findings with the Hubble Space telescope [Click on image for a bigger size]

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The spherical halo radius is at least 5 times the visible M 31 disc radius, probably much more than that!

Next, here is a very interesting article from the Hubble site, where the individual halo stars have been directly observed with Hubble’s Advanced Camera for Surveys.

You can read there also about some of the tricks used to separate Milky Way foreground stars from genuine halo stars bound by Andromeda's gravity!

In the first image below, I display their labled wide-field (ground-based) view of the Andromeda galaxy, showing the location of four fields where the NASA/ESA Hubble Space Telescope has been used to study a wide variety of Andromeda-based stars. Fields C and D are in the M 31 halo, while A refers to the disc.

[Click on image for a bigger size]

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Here is the image of M 31 halo stars from halo region D. Due to the enormously high resolution, one may look through the halo and spot other galaxies behind M 31.

[Click on image for a bigger size]

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For all 10000+ galaxies of celestia.Sci, I have used the relevant original scientific papers to implement halo stars precisely following their observations and analyses. A conspicuous population of the halo of M 31 consists of red giant stars that you will also be able to spot in my recent celestia.Sci rendering.

Unfortunately, I am only able to read some of those papers via my laboratory's (expensive) subscription. Naturally, I am not allowed to give them out of hand.

Before I will be able to quote the free scientific papers, I need to do another check. You'll be able to consult the free scientific papers very soon.

Lots of valuable detail is contained in these papers: the authors fit the surface brightness distribution of the halo stars with a inverse powerlaw 1/ r^2.2 out to projected distances r ~ 175 kpc = 570 500 LY from M 31's center!! Please recall that pure Newtonian gravity would lead to a 1/r^2 decrease (neglecting Dark Matter etc).

In order to render the halo stars in celestia.Sci, I made use of all advanced statistical star generation methods that I explained earlier in this celestia.Sci Development forum.

Soon, in a second part of this report, I'll expose some more details.

Finally, here is a celestia.Sci screenshot of the tightly wound Sa-type NGC 38 with its star halo and a bright foreground star. So you may get a feel already what the new feature looks like:

Following the literature, I have assumed for simplicity that the stellar halo properties measured explicitly for the Andromeda galaxy (M 31) are approximately universal throughout. Moreover (in a 3rd part), I use a supposedly universal stellar Luminosity function as extracted from nearby Milkyway stars (including HIPPARCOS stars) and as parametrized in E) above.

The shape of the skyplane projected halo is essentially circular, corresponding to a spherical shape in 3D. The authors find that the following inverse power law function fits the measured surface brightness of the halo as function of the projected radial distance R from the center:

Here , with R* = 30 kpc and a "core radius" parameter a_h = 5.2 kpc. From the analysis, a surprisingly large halo radius R_halo >= 175 kpc (2/3 of M 31's virial radius!) is deduced. The power index α was determined as α = 1.1 ± 0.1 which is very close to a drop off expected from pure gravity ~ 1/R^2, i.e. α = 1.0.

Here is a summary display of halo measurements and a display of the above fit from Ref 2 (arXiv, Fig. 10).

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Note: the data points are color-coded by the field's position angle, counted from M 31's minor axis. It is evident that the data is consistent with a circular shape of the skyplane projected halo!

The next important step is to deproject the observed circular skyplane projected surface brightness distribution into a proper distribution referring to 3D space, by following the exposed mathematics in B) above.

The constant R* = 30 kpc above does not enter the present discussion, since we can equivalently rewrite the fit function somewhat simplified as

in terms of a new reference flux constant i0.

As a quick orientation about the geometries involved, let me recall my drawing from the general discussion B):

[click on image for a bigger size]

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Keep in mind that R and r denote the projected and true halo radius, respectively, as apparent from the figure!

According to B), the radial behaviour of the halo's 3D surface brightness is obtained as solution of an Abel integral equation that takes a simple integral form in terms of the fitted 2D projected expression i(R):

with

The result reduces again to a simple inverse power law that decreases a bit faster, however, like 1/r^(2*alpha+1) for large r! Note the Γ function being a generalization of the familiar factorial n! = 1*2*3*....*n to non-integer argument.

Next, it is convenient to introduce 2 new parameters and a dimensionless, fractionalradius η via the transformation

Obviously, η takes values in (0,1). The second dimensionless parameter ratioexpresses the relative central core size in terms of a ratio of the core radius and the total halo radius r_halo.

The result now takes the simpler form

As it should be, the 3D surface brightness is seen to vanish for η = 1 or equivalently for r = r_halo.

Random Halo Star Generation (see C, D) above)

While in 3D, the random generation of the isotropic angular dependence involves of course just a uniform (i.e. constant!) Probability Density Function (PDF) (see C) above), the task to properly generate the star density as function of the distance r from center is a much less trivial affair (see D) above).

For the random halo star generation we conveniently use spherical polar coordinates and express the differential volume element dV in this familiar form

dV= r^2 dr dΏ = r_halo^3 * η^2 dη dΏ

For the random generation of the isotropic angular distribution, we again take a look at the differential solid angle element dΏ as in C) above

Note first of all, that our deprojected radial surface brightness distribution i_3D above has the meaning of a volume density (i.e. [#stars / volume]!). Thus, in order to normalize its integral to 1, as needed for a proper PDF, we now need to integrate over the spherical halo volume, like so

Note the resulting factor of η^2 which will suppress the halo star density towards small η and the factor 4π from the trivial integration over the solid angle.

Hence the properly normalized halo star density that will serve as our PDF, takes the following form

Since the normalization integral over i_3D is rather long, I displayed it in unevaluated form. Of course (by construction),

Here is a plot of our PDF that may well be more instructive:

You clearly see that our PDF with its strong depletion towards η -> 0, its sharp peak and its vanishing for η ->1 is very far from a constant, uniform PDF. This is why need to use a combination of the appropriate two methods described in D).

The first step consists in finding a suitable majorant function, simple enough in order to allow using the Inverse Transform Method described in D).

Remember: The majorant function needs also to be normalized to a unit area (just like the exact PDF), but after multiplication with a suitable constant C has to enclose our exact PDF entirely! The closer the shape of the majorant is to the exact PDF, the higher the efficiency!

In our case, finding the best majorant with these properties is relatively easy. Apart from a constant normalization factor, it looks like so:

where the majorant property is based on the obvious inequality:

and thus (for α > 0, w>0)

By comparing the exact PDF with this majorant, you now can prove the majorant property in 2 lines . Indeed, here is a comparison plot which clearly illustrates the claim:

Look at the legend below the plot. The constant C = 1.214911633 > 1and correspondingly we get a very high efficiency (100/C)

Next:i.e. for our case

resulting in a sufficiently simple elementary function on the right!Therefore, we may easily solve the inversion equation (see D) above!) U(η) = u for η with the result

This equation based on the majorant PDF allows ingenously to use uniform random values u in (0,1) as input from your built-in random generator routine and get output values η in (0,1) that are distributed according to our majorant function. Since our majorant was already pretty close in shape to the exact PDF, we only need to apply relatively small subsequent corrections to the generated sample of η values by means of the Acceptance - Rejection Method by John Von Neumann discussed in D).

All it takes is to throw another uniform random value uu in (0,1) and compare it with theratio

if uu <= f_by_ch accept this particular η, else reject η and try with a new η!

+++++++++++++++++++++++Here is now the final plot of success. I prepared a histogram of binned η values of 73728 halo stars sampled from the .Sci code by means of the described algorithm. Then I overlaid the previous plot with the properly normalized C * majorant (red) and our exact PDF (green). The histogram shape and normalization agrees perfectly with the exact (green) PDF for all η in (0,1)!+++++++++++++++++++++++

This quite long discussion has so far just covered the random generation of halo star positions in 3D space. There is another long section to come (after some rest ) that deals with the random generation of the halo star magnitudes and colors, in agreement with observation!

Please let me know whether the presented level was hopelessly too high, or whether people with some math/astrophysics background were getting out something from this report. The material is certainly not meant as a Newbie tutorial . It rather should provide a concise report about some scientific-level aspects of celestia.Sci.

For those of you who suffered through my previous long discussion about the implementation of galactic star halos into .Sci, here are just three typical examples of how galaxies now look together with their star halos.

I cut out a popular laptop image size of 1366 x 768 from the screenshots taken with my bigger, high-quality 1920 x 1200 monitor.

Always click on the images and then use browser full-screen (hit F11 in Firefox!)

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